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A178979 Triangular array read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the shortest block has length k (1<=k<=n). 1
1, 1, 1, 4, 0, 1, 11, 3, 0, 1, 41, 10, 0, 0, 1, 162, 30, 10, 0, 0, 1, 715, 126, 35, 0, 0, 0, 1, 3425, 623, 56, 35, 0, 0, 0, 1, 17722, 2934, 364, 126, 0, 0, 0, 0, 1, 98253, 15165, 2220, 210, 126, 0, 0, 0, 0, 1, 580317, 86900, 10560, 330, 462, 0, 0, 0, 0, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Row sums are Bell numbers A000110.

Column 1 is A000296 (shifted).

- Contributions:

Sum_{k>1} T(n,k) = A000296(n) count the set partitions with blocks of size > 1.

T(n,1) = A000296(n-1) count the set partitions with blocks of size = 1. Thus for the Bell numbers A000110(n) = Sum_{k>=1} T(n,k) = A000296(n-1) + A000296(n). - Peter Luschny, Apr 05 2011

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

Peter Luschny, Set partitions

FORMULA

E.g.f. for column k: exp((exp(x) - Sum_{i=0..k-1} x^i/i!)) - exp((exp(x) - Sum_{i=0..k} x^i/i!)).

EXAMPLE

T(4,2) = card ({12|34, 13|24, 14|23}) = 3. - Peter Luschny, Apr 05 2011

Triangle begins:

1

1,     1

4,     0,  1

11,    3,  0,  1

41,   10,  0,  0,  1

162,  30, 10,  0,  0,  1

715, 126, 35,  0,  0,  0,  1

MAPLE

g := k-> exp(x)*(1-(GAMMA(k, x)/GAMMA(k))); egf := k-> exp(g(k))-exp(g(k+1));

T := (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):

seq(seq(T(n, k), k=1..n), n=1..9); # Peter Luschny, Apr 05 2011

# second Maple program:

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,

       add(b(n-i*j, i+1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))

    end:

T:= (n, k)-> b(n, k) -b(n, k+1):

seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Mar 25 2016

MATHEMATICA

a[k_]:= Exp[x]-Sum[x^i/i!, {i, 0, k}]; Transpose[Table[Range[20]! Rest[CoefficientList[Series[Exp[a[k-1]]-Exp[a[k]], {x, 0, 20}], x]], {k, 1, 9}]]//Grid

CROSSREFS

Cf. A145877, A000110, A000296.

Sequence in context: A121301 A059056 A127153 * A228270 A266488 A189355

Adjacent sequences:  A178976 A178977 A178978 * A178980 A178981 A178982

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, Jan 02 2011

STATUS

approved

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Last modified July 22 00:05 EDT 2018. Contains 312888 sequences. (Running on oeis4.)